Export actions(animations) to FBX and open in Unity
I have the answer now:

The first problem is, that I don't clear the old actions when I start the script twice. Therefore, it appends the new actions to the old ones.
Now in Blender there are only the needed actions.

However, when I Export it to FBX and open it in Unity I have the same problem. 

You have to push the action down to a new strip:
Open the NLA-Editor and push that button next to your action.



For the export you have to select the following:



When you open your FBX in Unity there is only one action for each object.

How you do it in python:

#Add NLA-Strip
bpy.context.area.type = "NLA_EDITOR"
bpy.ops.nla.action_pushdown(channel_index = 1)#choose your index
#Export Settings 
bpy.ops.export_scene.fbx(filepath = yourfilepath,
                         use_selection=True,
                         bake_anim_use_all_bones = False,
                         bake_anim_use_nla_strips=True,
                         bake_anim_use_all_actions=False
                         )